We incorporate the behaviour of tax evasion into the standard two-dimensional Ising model and augment it by providing policy-makers with the opportunity to curb tax evasion via an appropriate enforcement mechanism. We discuss different network structures in which tax evasion may vary greatly over time if no measures of control are taken. Furthermore, we show that even minimal enforcement levels may help to alleviate this problem substantially.

On directed Barabási-Albert networks with two and seven neighbours selected by each added site, the Ising model was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. On these networks the Majority-vote model with noise is now studied through Monte Carlo simulations. However, in this model, the order-disorder phase transition of the order parameter is well defined in this system. We calculate the value of the critical noise parameter q c for several values of connectivity z of the directed Barabási-Albert network. The critical exponentes β/ν, γ/ν and 1/ν were calculated for several values of z.Keywords:Monte Carlo simulation,vote , networks, nonequilibrium. IntroductionIt has been argued that nonequilibrium stochastic spin systems on regular square lattice with up-down symmetry fall in the universality class of the equilibrium Ising model [1]. This conjecture was found in several models that do not obey detailed balance [2,3,4]. Campos et al.[5] investigated the majority-vote model on small-world network by rewiring the two dimensional square lattice. These small-world networks, aside from presenting quenched disorder, also posses long-range interactions. They found that the critical exponents γ/ν and β/ν are different from the Ising model and depend on the rewiring probability. However, it was not evident that the exponent change was due to the disordered nature of the network or due to the presence of long-range interactions. Lima et al.[6] studied the majority-vote model on Voronoi-Delaunay random lattices with periodic boundary conditions. These lattices posses natural quenched disorder in their conecctions. They showed that presence of quenched connectivity disorder is enough to alter the exponents β/ν and γ/ν from the pure model and therefore that is a relevant term to such non-equilibrium phase-transition. Sumour and Shabat [7,8]

The stationary critical properties of the isotropic majority vote model on random lattices with quenched connectivity disorder are calculated by using Monte Carlo simulations and finite size analysis. The critical exponents γ and β are found to be different from those of the Ising and majority vote on the square lattice model and the critical noise parameter is found to be qc = 0.117 ± 0.005.

Through Monte Carlo Simulation, the well-known majority-vote model has been studied with noise on directed random graphs. In order to characterize completely the observed order-disorder phase transition, the critical noise parameter q c , as well as the critical exponents β/ν, γ/ν and 1/ν have been calculated as a function of the connectivity z of the random graph.

On directed lattices, with half as many neighbours as in the usual undirected lattices, the Ising model does not seem to show a spontaneous magnetisation, at least for lower dimensions. Instead, the decay time for flipping of the magnetisation follows an Arrhenius law on the square and simple cubic lattice. On directed Barabási-Albert networks with two and seven neighbours selected by each added site, Metropolis and Glauber algorithms give similar results, while for Wolff cluster flipping the magnetisation decays exponentially with time. IntroductionSánchez et al some years ago showed that the Metropolis algorithm, applied to directed Watts-Strogatz (small-world) networks has a spontaneous magnetisation, even in the limit of directed random graphs [1]. More recently, Sumour and Shabat [2,3] investigated Ising models on directed Barabási-Albert networks [4] with the usual Glauber dynamics. No spontaneous magnetisation was found, in contrast to the case of undirected Barabási-Albert networks [5,7] where a spontaneous magnetisation was found below a critical temperature which increases logarithmically with system size. Now we simulate directed square, cubic and hypercubic lattices in two to five dimensions with heat bath dynamics in order to separate the network effects form the effects of directedness. And we compare different spin flip algorithms, including cluster flips [8], for Ising-Barabási-Albert networks. In all these cases spins were flipped according to algorithms well established to equilibrate systems described by an Ising energy, even though the directed systems are not described by such an energy [1], in contrast to an earlier assertion by one of us [3].

On (3, 12 2 ), (4,6,12) and (4, 8 2 ) Archimedean lattices, the critical properties of majority-vote model are considered and studied using the Glauber transition rate proposed by Kwak et all. [Phys. Rev. E, 75, 061110 (2007)] rather than the traditional majority-vote with noise [José Mário de Oliveira, J. Stat. Phys. 66, 273 (1992)]. The critical temperature and the critical exponents for this transition rate are obtained from extensive Monte Carlo simulations and with a finite size scaling analysis. The calculated values of the critical temperatures Binder cumulant are Tc = 0.363(2) and U * 4 = 0.577(4); Tc = 0.651(3) and U * 4 = 0.612(5); and Tc = 0.667(2) and U * 4 = 0.613(5) for (3, 12 2 ), (4, 6, 12) and (4, 8 2 ) lattices, respectively. The critical exponents β/ν, γ/ν and 1/ν for this model are 0.237(6), 0.73(10), and 0.83(5); 0.105(8), 1.28(11), and 1.16(5); 0.113(2), 1.60(4), and 0.84(6) for (3, 12 2 ), (4, 6, 12) and (4, 8 2 ) lattices, respectively. These results differ from the usual Ising model results and the majority-vote model on so-far studied regular lattices or complex networks.

On directed Small-World networks the Majority-vote model with noise is now studied through Monte Carlo simulations. In this model, the orderdisorder phase transition of the order parameter is well defined in this system. We calculate the value of the critical noise parameter q c for several values of rewiring probability p of the directed Small-World network. The critical exponentes β/ν, γ/ν and 1/ν were calculated for several values of p.

Here, the model of non-equilibrium model with two states ($-1,+1$) and a noise $q$ on simple square lattices proposed for M.J. Oliveira (1992) following the conjecture of up-down symmetry of Grinstein and colleagues (1985) is studied and generalized. This model is well-known, today, as Majority-Vote Model. They showed, through Monte Carlo simulations, that their obtained results fall into the universality class of the equilibrium Ising model on a square lattice. In this work, we generalize the Majority-Vote Model for a version with three states, now including the zero state, ($-1,0,+1$) in two dimensions. Using Monte Carlo simulations, we showed that our model falls into the universality class of the spin-1 ($-1,0,+1$) and spin-1/2 Ising model and also agree with Majority-Vote Model proposed for M.J. Oliveira (1992) . The exponents ratio obtained for our model was $\gamma/\nu =1.77(3)$, $\beta/\nu=0.121(5)$, and $1/\nu =1.03(5)$. The critical noise obtained and the fourth-order cumulant were $q_{c}=0.106(5)$ and $U^{*}=0.62(3)$.Comment: 13 pages, 6 figure